Equal masses of methane and oxygen are mixed in an empty container...

Let the mass of methane and oxygen is $w$$\begin{aligned} & \text{ mole fraction of oxygen }=\frac{\frac{w}{32}}{\frac{w}{32}+\frac{w}{16}} \\ & =\frac{\frac{1}{32}}{\frac{1}{32}+\frac{1}{16}}=\frac{\frac{1}{32}}{\frac{3}{32}}=\frac{1}{3} \end{aligned}$$$Let the total pressure be$\mathrm{P}$The pressure exerted by oxygen (partial pressure)$=\mathrm{X}_{\mathrm{O}_2} \times \mathrm{P}_{\text{total }} \Rightarrow \mathrm{P} \times \frac{1}{3}$ Hence, (C) is correct.